Maximal Regularity for Integral Equations in Banach Spaces

Abstract

We study maximal regularity in periodic Besov spaces $B_{p,q}^s(\mathbb T, X)$ for the integral equations ($P$): $u(t) = A \int^{t}_{-\infty} a(t-s) u(s) ds) + B \int^{t}_{-\infty} b(t-s) u(s) ds + f(t)$ on $[0,2\pi]$ with periodic boundary condition $u(0) = u(2\pi)$, where $A$ and $B$ are closed operators in a Banach space $X$, $a, \ b \in L^1(\mathbb R_+)$ and $f$ is a given function defined on $[0,2\pi]$ with values in $X$. Under suitable assumptions on the kernels $a, \ b$ and the closed operators $A, \ B$, we completely characterize $B_{p,q}^s$-maximal regularity of ($P$).